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Answer:
Step-by-step explanation:
Hello!
The variance and standard deviation of a distribution are measurements of dispersion. They show the degree of variation of the values of a dataset regarding the mean.
<u>Variance:</u>
The variance is defined as the mean of the squares of the difference between the values of the variable and the mean, symbolically:
V(X)= S²= E(X-X[bar])²= <u> </u><u>∑(</u>
<u>-X[bar])² </u>
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<u>Properties:</u>
1. The variance is non-negative:
V(X) ≥ 0
Since it is calculated as the summation of squared values it is not possible for it to be negative.
2. The variance of a constant (k) is zero:
V(k)= E(k-k)²= 0
3. The variance is unaffected by changes in the parameters of position, so if for example, a constant is added to the variable X+k, since it is added to all values of the variable, the variance remains unchanged:
V(X+k)= V(X)
Same happens if a constant is subtracted from the variable:
V(X-k)= V(X)
4. If you multiply the variable X by a constant k, the resulting variance will be the variance of the variable by the square of the constant:
V(X*k)= k² * V(X)
5. If you add or subtract two independent variables, the resulting variance for both cases will be the summation of their variances:
V(X±Y)= V(X)+V(Y)
When you add or subtract two dependent variables, the variance of the resulting variable will be the summation of both variances minus their covariance:
V(X±Y)= V(X)+V(Y)-Cov(XY)
<u><em>Note:</em></u><em> covariance is the measure of joint variability of two random variables. </em>
<u>Standard deviation:</u>
The standard deviation is defined as the square root of the variance. The properties of the variance apply to it too.
This measurement is often chosen over the variance since is easier to express the dispersion of a variable over non-square units or values.
S= √S²
I hope this helps!